In: Found. Physics., Vol. 23, No. 3, (1993) 365387.

Zitterbewegung Modeling

David Hestenes

Abstract. Guidelines for constructing point particle models of the elec-tron with zitterbewegung and other features of the Dirac theory are dis-cussed. Such models may at least be useful approximations to the Diractheory, but the more exciting possibility is that this approach may lead toa more fundamental reality.

1. INTRODUCTION

For many years I mulled over a variation of Schrodingers zitterbewegung concept toaccount for some of the most perplexing features of quantum theory. I was reluctant topublish my ideas, however, because the supporting arguments were mainly qualitative, andphysics tradition demands a quantitative formulation which can be subjected to experi-mental test. Unfortunately, the road to a quantitative version appeared to be too long anddicult to traverse without help from my colleagues. But how could I enlist their helpwithout publishing my qualitative arguments to get their attention?

The impetus to break this impasse came from the inspring example of Asim Barut.Barut is unusual among theoretical physicists, not only for the rich variety of clever ideashe generates, but also for publishing theoretical fragments which are sometimes mutuallyincompatible. Though each of his papers is internally coherent, he is not afraid to contradicthimself from one paper to the next or try out alternative theoretical styles. He rapidlyexplores one idea after another in print, searching, I suppose, for ever deeper insights.Consequently, his papers are an exceptionally rich source of new ideas, arguments, andviewpoints. It was Baruts inspiring example that convinced me nally to publish myqualitative analysis of the zitterbewegung in Ref. 1. Despite its defects, that publicationdid, indeed, stimulate helpful interaction with my peers and led to a mathematically well-grounded zitterbewegung interpretation of the Dirac theory [2] which is the starting pointfor the present paper. In appreciation for his positive inuence on these events, I dedicatethis paper to Asim Barut.

The main motivation for analyzing zitterbewegung (ZBW) models is to explain the elec-trons spin S and magnetic moment as generated by a local circulation of mass and charge.Experimental evidence rules out the possibility that the electron is an extended body, forthe relativistic limitation on velocity implies that, to generate S and , the dimensionsof the body cannot be less than a Compton wavelength ( 1013 m), whereas scatteringexperiments show that the electron cannot be larger than 1016 m. This leaves open thepossibility that the electron can be regarded as a point charge which generates S and byan inherent local circular motion.

For a particle moving in a circle at the speed of light c, the radius of the circle is relatedto the circular frequency by

= c = 1 (1.1)

1

If the particle is presumed to have mass m and generate the electron spin by this motion,then

|S | = 12 h = m2 (1.2)Thus, radius (wavelength), mass, and frequency are all related by

=h

2m=

1

(1.3)

This may appear to be a hopelessly simple-minded explanation for electron spin, but (sur-prise!!) it has been shown to be completely consistent with the mathematical structure ofthe Dirac theory.(2) Thus, despite the appearance of electron mass in the Dirac equation,the electron may indeed be moving with the speed of light. Contrary to long-standingclassical arguments, such motion can produce a gyromagnetic ratio of 2. Moreover, thecomplex phase factor can be interpreted as a direct representation of the circular ZBW.Thus, the phase factor

ei/h = ei/2 (1.4)

is associated with the frequency = , where dierentiation is with respect to the propertime on the worldline of the center of curvature for the circular ZBW. Note that (1.4) is ahalf-angle representation of the circle. Comparison of (1.4) with (1.3) gives

= 12 h = m (1.5)

showing that m must be a variable mass. It is equal to the electron rest mass m0 onlyfor a free particle. This interpretation of the phase is derived from the standard energy-momentum operator

p = ih eA (1.6)Operating on the phase factor (1.4), we obtain the gauge-invariant energy-momentum vector

p = eA (1.7)

Contraction with the Dirac velocity v (for the center of curvature) gives

p v = eA v (1.8)

For m = p v this diers from (1.5), an important point to be discussed later.The compatibility of this ZBW interpretation with the details of the Dirac theory is

demonstrated in Ref. 2. Its very success, however, suggests the possibility of a deeper theoryof electrons. It suggest that the Dirac theory actually describes a statistical ensemble ofpossible electron motions. If that is correct, it should be possible to nd equations of motionfor a single electron history and then derive the Dirac theory by statistical arguments. Thatpossibility is explored in this paper, where denite equations of motion for a single pointparticle with ZBW are written down and discussed. With the ZBW interpretation as aguide, the equations are designed to capture the essential features of the Dirac equation.

Of course, such working backward from presumed statistical averages to underlyingdynamics is a guessing game, prone to error in its early stages. For that reason, this papermakes no pretense of producing a denitive ZBW model. The objective, rather, is

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(1) to motivate and describe both kinematical and dynamical aspects of the ZBW concept,

(2) to formulate and implement basic design principles for ZBW modeling, and

(3) to identify and sharpen issues that must be resolved to produce a fully viable theory.

This is not the place to discuss experimental tests of the ZBW.If the resulting equations do not in fact describe a deeper reality, at least they may

constitute a useful approximation to the Dirac theory, a new version of semiclassicalapproximations which have already proved their value.

2. OBSERVABLES OF THE DIRAC THEORY

The present paper is a continuation of work published in this journal,(2) so some famil-iarity with that work can be presumed as background. In particular, spacetime algebra willbe employed as the essential mathematical tool. The rudiments of spacetime algebra areexplained in Ref. 2 and expanded in many references cited therein. A basic feature of thealgebra is that the four in the Dirac theory are not regarded as matrices but as vectorsconstituting a xed orthonormal frame for spacetime.

As a guide for the design of a ZBW model, the formulation of Dirac observables in termsof spacetime algebra is reviewed here. The Dirac wave function for a electron can be writtenin the form

= (ei)1/2R (2.1)

where and are scalars, i = 0123 is the unit pseudoscalar, and R is a unimodularspinor, that is, an even multivector satisfying

RR = 1 (2.2)

The form (2.1 ) can be interpreted physically as a factorization of the electron wave functioninto a statistical factor (ei)1/2 and a kinematical factor R. Thus, the scalar modulus is interpreted as a probability density, although the interpretation of raises problems tobe discussed later.

The kinematical factor R = R(x) determines an orthonormal frame of vectors e = e(x)at each spacetime point x by

e = RR (2.3)

This frame describes the kinematics of electron motion. The Dirac current

J = 0 = e0 (2.4)

has vanishing divergence, so it determines a family of electron streamlines with tangentvector

v =dx

d= x. = e0 (2.5)

The vector eld v = v(x) can be interpreted statistically as a prediction of the electronvelocity at each spacetime point x. The electron spin (or polarization) vector is dened by

s = 12 he3 (2.6)

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However, the electron spin is more properly characterized as a bivector quantity S denedby

S = 12 he2e1 = isv (2.7)

The unimodularity condition (2.2) implies that the derivative of R in the direction can be written in the form

R = 12R (2.8)

where is a bivector quantity. Consequently, the derivatives of (2.3) can be written

e = e (2.9)

showing that is to be interpreted as the rotational velocity of the frame {e} whendisplaced in the direction.

A physical meaning is assigned to the by comparing them to the spin S. Write

S = S + S + S = P + iq + S (2.10)

where

P = P = Sq = q = i( S) = (iS) = (sv)

= v (s) = s (v)S = S (2.13)

(2.11)

(2.12)

In the Dirac theory, the vector P in (2.11) is implicitly interpreted as the canonical (energy)momentum, which is related to the kinetic (energy) momentum vector p by

P = p + eA (2.14)

where A = A is the electromagnetic vector potential and e is the electron charge. Thus,Eq. (2.11) explicitly attributes the energy-momentum of the electron to the rotation ratein the plane of the spin S. This is a key to the ZBW interpretation, and identies (2.11)as the full generalization of (1.7).

Along an electron streamline, the rotational velocity is

= v (2.15)

where v = v , so (2.8) yieldsR = 12R (2.16)

where the overdot indicates dierentiation with respect to proper time along the streamline.The motion of the e, along a streamline is therefore described by

e = e (2.17)

In particular,v = v (2.18)

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becomes an equation of motion for the streamline when = (x) is specied. And (2.13)yields

S = S (2.19)which determines the spin precession along a streamline. Thus, the dynamics of electronmotion can be reduced to determining , a matter to be discussed later.

Along a streamline, (2.10) and (2.12) yield

S = S + S + S = v P + S + i(s v) (2.20)

This summarizes the relation of to observables. Indeed, it can be solved for by multi-plication with S1.

Furthermore, (2.11) and (2.14) yield

m = p v = S eA v (2.21)

which denes a variable mass m for the electron in accordance with the remarks in theintroduction.

All the above relations are implicit in the standard Dirac theory and do not involve anyapproximation, so they are equally applicable to the physical interpretation of any solutionof the Dirac equation. The zitterbewegung interpretation requires one further physicalassumption which, however, does not alter the mathematical structure of the Dirac theory.To account for the spin S as generated by electron motion, the electron must have acomponent of velocity in the spin plane. This is readily achieved by identifying the electronvelocity with the null vector

u = e0 e2 (2.22)From (2.17) it follows that, for specied = (x), it has the equation of motion

u = u (2.23)

Its solutions are always lightlike helixes winding around the electron streamlines with ve-locity v = e0.

3. ZITTERBEWEGUNG KINEMATICS

Now we get to the main business of this article: to construct and analyze point par-ticle models of the electron consistent with the features of the Dirac theory described inthe preceding section. The rst step is to dene the appropriate kinematic variables andconstraints.

The electron is presumed to have a lightlike history z = z() in spacetime with velocity

u =dz

d= z , so u2 = 0 (3.1)

A proper time cannot be dened on such a curve, so an additional condition is neededbelow to specify the parameter uniquely.

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The electron motion is assumed to possess an intrinsic angular momentum, or spin,S = S(), a spacelike bivector with constant magnitude specied by

S2 = h2

4(3.2)

The velocity u can be decomposed into a component v orthogonal to S and a componentr in the S-plane dened by

v = (u S)S1 (3.3)r = (u S)S1 (3.4)

Note that v and r are merely auxiliary variables dened by these equations for purposes ofanalysis. This determines the decomposition

u = v + r (3.5)

withv r = 0 (3.6)

and, by (3.1),v2 = r2 (3.7)

A scaling for the parameter is determined by assuming that

v2 = v u = (u S)2

S2= 1 (3.8)

It will be seen later, though, that this is a mere convention, and not really a physicalconstraint on the model.

Equation (3.5) suggests the decomposition

z() = x() + r() (3.9)

withv = x. (3.10)

This expresses the electron history z() as a circulating motion with radius vector r(),angular momentum S, and a center of curvature with a timelike history x = x().

All these features of our particle model correspond exactly to kinematic features of theDirac theory. Indeed, (3.5) is the same as (2.22) with v = e0 and r = e2. It followsthat the denition of a comoving frame by (2.3) and the spinor equation of motion (2.8),with their sundry implications (2.17), (2.18), (2.19), and (2.22), apply also to the particlemodel, the sole dierence being that all quantities are dened only on the particle historyz() rather than as elds distributed over spacetime. As in the Dirac theory, the dynamicsin those equations are entirely determined by specifying the rotational velocity = (z()),a matter to be discussed in the next section.

The situation is dierent, however, for momentum and mass. We dene the momentump in a standard way by writing

p = mu = mv + mr (3.11)

6

This cannot be identied with the momentum p dened in the Dirac theory by (2.11) and(2.14), because it has the rapidly uctuating component mr. Rather, it will be argued laterthat the Dirac momentum corresponds to the average of (3.11) over a ZBW period.

The mass m in (3.11) remains to be defned. In accordance with the ZBW concept, weassume that it has the same origin as the spin, following (2.21) in dening it in relation to(3.11) by

m = S = p v = p r (3.12)This may dier from (2.21), however, due to averaging over a ZBW period. The essentialkinematic specications for our model are now complete, but it will be convenient to relatemass and momentum to spin by writing

S = mr r (3.13)This amounts to a denition of the vector radius of curvature. Substitution into (3.12)yields

(r r) = 1 (3.14)This generalizes Eq. (1.1), where

= S = 2h S (3.15)

is the ZBW (circular) frequency.The mass m, the scalar radius of curvature = | r |, and the ZBW frequency all covary

with changes in . To deduce how thay are interrelated, note that by (2.17), r = e2implies that

..r = r (3.16)

which, in turn, implies..r r = 0 (3.17)

Also, r2 = | r |2 = 2 implies thatr r = (3.18)

From (3.13) and (2.19), we obtain

S = mr r + m..r r = S = m( r) r + mr ( r)Whence,

mr r = mr ( r) (3.19)Multiplying this by r r and using

(r r)2 = h2

2m2(3.20)

obtained from (3.2), we nd

m

(h24m2

)= m(r r)( r) r = m

7

Therefore,h28

m

m3=

d

d

(h2

4m2

)=

d2

d(3.21)

Setting the integration constant to zero, we thus obtain

m = 12 h (3.22)

exactly as specied by (1.3). Here, however, it is clear that m and can be variable, withderivatives related by

m

m=

(3.23)

It should be evident that we have a new concept of mass here, through, to a certainextent, it was already implicit in the Dirac theory. The vague concept of mass as some kindof material stu is completely gone. No longer is vanishing mass a distinguishing featureof particles moving with the speed of light, because p2 = m2u2 = 0 here.

The relation m = p v expresses the inertial property of mass as a relation betweenmomentum and velocity. This relation conforms to the relativistic concept of mass as ameasure of energy content, but that just means that mass and energy are essentially one andthe same. The complementary relation m = S = 12 h asserts that mass is a frequencymeasure. This conforms to de Broglies original idea that the electron contains an internalclock with frequency determined by its mass, though for a free particle the ZBW frequencydiers from the de Broglie frequency by a factor of 2, and the mass varies with interactionsin the present model. Moreover, the relation m = 12 h says that this frequency measuresthe radius of curvature of the electron worldline, so it is a thoroughly geometrical quantity.All this suggests that the mass relates our externally imposed time scale to a time scaleintrinsic to the electron.

Considerations in the next section suggest that mass is a measure of self-interaction. Inother words, mass measures the coupling strength of the electron to its own eld. This,then, is the source of the inertial property of electron mass. Intuitively speaking, whenaccelerated by external elds, the electron must drag its own eld along.

The direct relation of mass to ZBW frequency described here comes from assumingm = S = , which diers from the relations (2.21) and (1.8) in the Dirac theory by theeA v term. This dierence was introduced into the particle model for a physical reason.The Dirac equation allows the value of to be changed by a gauge transformation, whereasthe ZBW interpretation of as an objective property of electron motion related to its spinimplies that should have a unique value. It will be argued later that the eA v term mayappear in the Dirac theory from time averaging which is inherent in the theory. On theother hand, if the eA v term is found to be essential to account for physical facts suchas the AharonovBohm eect, it could be incorporated into a particle model. This is aphysical issue to be resolved by further theoretical and experimental analysis.

4. ZITTERBEWEGUNG DYNAMICS

As already mentioned, the dynamics of electron motion is completely determined byspecifying the rotational velocity = (z()). An explicit expression for in the Dirac

8

theory was derived in Ref. 3 and is discussed in Section 6. With the Dirac as a guide,our problem is to guess a suitable expression for in our particle model. The simplestpossibility to consider is

=e

m0F + m0S1 (4.1)

where m0 is the electron rest mass and the bivector F is any specied external electro-magnetic eld. This determines a well-dened dynamical model which can account for asignicant range of physical phenomena.

Substituting (4.1) into (2.18) and using v S = 0, we obtain

m0v = eF v (4.2)

which is exactly the form of the classical Lorentz force, so it is evident that classical elec-trodynamics can be recovered from the model. However, if (4.2) is to be regarded as anequation of motion for the center of curvature x = x() with v = x., it is necessary toexpress F as a function of x instead of z. This is most naturally done by means of therst-order Taylor expansion.

F (z) = F (x + r) = F (x) + r F (x) (4.3)

where r = r,. Thus, the classical result will be obtained if and only if the last termin (4.3) in negligible compared to F (x). Let us call this case the classical approximation.Of course, the last term in (4.3) is a ZBW eect.

Substituting (4.1) into (2.19) yields

S =e

m0F S (4.4)

which exhibits the g = 2 value for the gyromagnetic ratio in the classical approximation,exactly as in the Dirac theory. Note that the ZBW correction on the right side of (4.3)vanishes when F is constant, so the classical approximation applies rigorously in that case,the case which has been of greatest interest in measurements of the g-factor.

While (4.2) determines the electrons center of curvature, and (4.4) determines the spinprecession along the worldline, the remaining feature of electron motion is the ZBW fre-quency determined by substituting (4.1) into (3.12) or (3.15), with the result

m =e

m0F S + m0 (4.5)

Thus, the external eld produces a gauge-invariant mass shift of Larmor type.Instead of trying to integrate the equations for v and S directly, it is advisable to employ

the spinor equation (2.16) which, for given by (4.1), becomes with the help of (2.2) and(2.7),

R =e

2m0FR +

2m0h

R21 (4.6)

This equation has solutions of the form

R = R0e21/h (4.7)

9

where R0 satises the reduced equation

R0 =e

2m0FR0 (4.8)

and the phase can be obtained by integrating (4.5) with m = after S has been foundby integrating (4.8).

Solutions of (4.8) in the classical approximation have been found in Ref. 4 for the caseswhere F is a constant, plane wave, or Coulomb eld, and the equation x. = v was integratedto nd explicit expressions for the worldline in each case.

The form of (4.1) suggests various generalizations, such as

=e

m0(F + Fs) (4.9)

where Fs is the electrons self-eld, that is, the electromagnetic eld of the electron itselfevaluated on the electrons worldline. The self-eld has the general form

e

m0Fs = m0S1 + ex +

e

m0FRR (4.10)

where FRR is the radiative reaction eld of the electron. Since radiative reaction is notincluded in the Dirac theory, we can defer discussion of the FRR term to another occasion.The remaining terms must then describe nonradiative eects of self-interaction.

The term m0S1 in (4.1) and (4.10), which is crucial for generating the electron restmass and spin, has the bivector form of a magnetic eld. This leads to the conclusion thatthe electrons electromagnetic selnteraction is fundamentally of magnetic type.

This also raises the question of what form the electrons eld takes at points away fromthe electron worldline. A major reason for developing a ZBW model in the rst placewas to explain the electrons magnetic dipole eld. If this explanation is taken seriously,then the dipole eld must be regarded as an average over a ZBW period, and the actualeld must also contain a high-frequency component Fzbw which oscillates with the ZBWfrequency. As argued in Refs. 1 and 2, this has the potential for explaining some of themost prominent and perplexing features of quantum theory, such as electron diractionand the Pauli principle, as resonant interactions mediated by the ZBW eld Fzbw. Notethat (4.1) can be employed to study this possibility without further modication. It isonly necessary to regard the external eld Fin (4.1) as including ZBW elds from externalsources. However, a quantitative analysis of ZBW resonances must be deferred to anothertime.

Of course, the existence of a stable uctuating ZBW eld accompanying even a free elec-tron raises questions about radiation which must be addressed eventually. The viewpointadopted here is that the Dirac theory suggests that such elds exist, and we should pushthe study of their putative eects on electron motion as far as possible before attackingthe fundamental problems of radiation and self-interaction. Equation (4.10) describes thecoupling of the electron to its own eld. Even for a free particle, the m0S1 term impliesthat the electrons momentum (3.11) has a component mr which uctuates (rotates ) withthe ZBW frequency. Presumably, this reects a uctuation of momentum in the electronsZBW eld. The nonspecic term ex is included in (4.10) to cover the possibility that the

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electrons eld may have nonradiating excitations which appear even in the motion of afree particle. An example will be given in the next section.

5. ZITTERBEWGUNG AVERAGES

It is of interest to eliminate ZBW oscillations from the equations of motion by averagingover a ZBW period for two good reasons at least. First, these high-frequency oscillations(on the order of 1021 Hz) are irrelevant in many situations (such as the classical limit), soit is advisable to systematically eliminate the need to consider them. Second, the (energy)momentum vector in the Dirac theory does not exhibit the ZBW uctuations of the mo-mentum vector p dened by (3.11), so it must be regarded as some kind of average p of p.Thus, the study of ZBW averages is essential if the Dirac theory is to be derived from anunderlying ZBW substructure.

Even apart from the complication that the ZBW period T = 2/ is a variable quantity,the denition of a ZBW average is not entirely straightforward, because it must be compati-ble with invariant constraints on ZBW kinematics. For example, for the center-of-curvaturevelocity the straightforward denition of an average v = v() by

v 1T

0

v()d =x() x(0)

T(5.1)

is unsatisfactory, because it fails to preserve the constraint v2 = 1. For any curved historyx(),

|x() x(0) | >

0

| dx | = T

implies v2 > 1. This defect could be corrected by introducing a parameter 0 and revisingthe denition (5.1) to

v cos0 =1T

0

v()d (5.2)

with v2 = 1 by denition. A parameter like 0 does indeed appear in the Dirac theory, butwe shall entertain other, probably deeper, reasons for that. Since v is orthogonal to theZBW uctuations, it is aected by them only indirectly through the equations of motion.It appears permissible, therefore, to take v = v by denition.

To maintain the constraint v S = 0, it is necessary to dene S = S also. In Ref. 2,I made the mistake of suggesting that v be dened by the ZBW average u . Instead, wetake v as dened already by (3.3) and dene p indirectly by imposing angular momentumconservation as a constraint. In Ref. 2, it was correctly argued that compatibility with theDirac theory requires that the average angular momentum J must satisfy the decompositioninto orbital and spin components dened by

J = p z = p x + S (5.3)

Of course, momentum conservation on the average requires that

p = f (5.4)

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where f is the average force on the particle. Angular momentum conservation then requiresthat

J = p x + p v + S = f x (5.5)Therefore,

S + p v = 0 (5.6)As noted in Ref. 3, this equation was rst formulated by Wessenho and has been studiedby many authors as a classical model for a particle with spin. Its appearance here gives itnew signicance as an approximation to the Dirac equation.

Dening the average mass bym = p v = p v (5.7)

and adding it to (5.6), we obtainvp = m + S (5.8)

This can be solved explicitly for

p = v(m + S) = mv + v S = mv + S v (5.9)

where the last equality comes from dierentiating v S = 0, and it is noted that (5.6)implies as well as

v S = 0 (5.10)as well as

p S = 0 (5.11)Consistency with the conservation laws thus implies that we must take (5.8) as the denitionof p. It will be seen in the next section that this corresponds to the Gordon decompositionin the Dirac theory. Comparison of (5.8) with (3.11) implies that

mr = v S (5.12)

an altogether reasonable result, which implies that p = mu = mv if and only if v S = 0.The mean dynamics is determined by assuming that it is governed by the average of

the rotational velocity over a ZBW period. This implies that

S = S (5.13)

for example. Accordingly, averaging (3.12) gives

m = S = p v (5.14)

so the analog of (2.20) in this case takes the form

S = m + S + i(s v) (5.15)

Whence, comparison with (5.8) gives

S = vp + i(s v) (5.16)

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The last term in (5.16) is purely kinematical, but (5.10) puts a constraint on s, for thederivative of is = Sv = S v is then is = S v = i(sv) v. Whence,

s = (s v)v = (s v)v (5.17)

which impliess v = 0 (5.18)

as well asv S = 0 (5.19)

An equation of motion for the mass can now be derived. From the right side of (5.9), itfollows that

p v = 0 (5.20)and (5.13) implies

S = ( S) = 0 (5.21)for any value of . Therefore, the derivative of (5.14) gives

m = S = v p = v f (5.22)

This is consistent with a general force law of the form

p = f = m v + ( S)v (5.23)

However, when radiation is ignored, f should presumably depend only on the external eldF in (4.9). Accordingly, since (4.3) implies

F (z) = F (x) (5.24)

for r = 0, (5.23) should be replaced by

p = f = mm0

eF v +(

e

m0S F

)(5.25)

which is related to (5.22) by

m = v f = v (

e

m0S F

)=

e

m0S F (5.26)

The force law (5.25) appears to be the most general expression for f which holds forarbitrary F and is consistent with the conditions on p. The mean equations of motionare thus determined by consistency conditions without a formal averaging procedure. Theratio m/m0 in (5.25) is strange, but a similar ratio appears in the Dirac theory. Of course,it may be that some modication of the expression for in (4.9) is needed.

Substituting (5.9) into (5.25) and using (5.26) as well as (5.19), we obtain

mv + v .. S = emm0

F v (5.27)

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This equation has a nontrivial solution even for F = 0, but it must be integrated togetherwith Eq. (5.13) for S. The general free particle solution is most easily obtained by deter-mining the constants of motion. Of course, p is one such constant. The bivector is alsoconstant if and only if s = 0, for in that case (5.16) yields

= vpS1 = is1p (5.28)

The spinor equation (2.16) then integrates to

R = e12 (5.29)

so the spin undergoes constant precession described by

S = RS0R (5.30)

The history x = x() is most easily found by writing (5.8) in the form

v = (m + S)/p (5.31)

which integrates immediately to

x() = x0 + mp1 + RS0p1R (5.32)

where the fact that p commutes with has been used. This solution is a timelike helixwith a precessing velocity

v = Rv0R (5.33)

What physical meaning can be ascribed to such a solution? Comparison with (4.10) suggeststhat

= m0S1 + ex (5.34)

As suggested in Section 4, the ex may describe self-interaction due to stable excitationsof the electrons ZBW eld. For the ground state self-interaction, ex = 0 which impliesS = 0 and m = m0.

Returning to the general problem of ZBW averaging, we note that the net eect of theaveraging is to change Eqs. (2.16) and (2.17) to

R = 12R (5.35)

ande = e (5.36)

This reduces the problem to determining . Evidently cannot be found by simpleaveraging because certain consistency constraints must be satised. Though a thoroughlyjustied derivation of is not possible at this time, comparison of (5.27) with (5.36) and(4.1) suggests that it should have the form

=e

m0F + m0S1 +

1m

..S (5.37)

14

This diers in form from Eq. (4.1) for only in the..S term. It seems, therefore, that the

..S

arises from the averaging, though the interpretation of (5.34) suggests that it may describean excitation of the ZBW eld and so should be included in even without averaging.

Aside from (possibly) altering the form of , the ZBW averaging changes the functionaldependence from = (z()) dened on the electron worldline z() to = (x())dened on the center-of-curvature worldline x() This has the eect of decoupling thecircular motion in the S-plane from the equations of motion for v and S, so S could beassigned any value without aecting v and S. Determining the average angular velocity (ormass) of the ZBW therefore requires special considerations. Though a denitive resolutionof the issue is not to be expected at this time, here is an important consideration: It maybe that the dierence between the ZBW mass-angle relation m = and the Dirac relation(1.8) is a consequence of averaging. This possibility can be expressed by rewriting (1.8) inthe form

m = p v = = eA v (5.38)To see how the A v term might be generated by averaging, consider the line integral overa ZBW period T = 2/,

T0

A dz = T

0

A dx + T

0

A dr (5.39)

where (3.9) has been employed to decompose it into two terms. Stokes theorem gives

T0

A dr =

(d2r) ( A) Thbar

S F (x) (5.40)

Whence, (5.39) yields1T

T0

A dz A v + S Fh

(5.41)

Thus, the line integral generates the gauge-invariant average of the S F mass shift termin (4.5), but at the expense of introducing the gauge-dependent term A v. Is the A vterm an artifact of the averaging process? If this term is really needed to account for theAharonhov-Bohm eect, as is widely believed, then the ZBW phase velocity must begauge dependent, so it does not have a unique value, which appears to be incompatiblewith its ZBW interpretation as a property of a unique world line. These remarks are aimedat sharpening the issue with no pretense of a nal resolution.

6. RELATION TO THE DIRAC THEORY

The Dirac equation can be reformulated as a kind of constitutive equation relating theobservables , v, S, p and dened in Section 2. This was done in Ref. 3, with the result

(p iq)ei = m0v (Sei) (6.1)

where = . The symbol p for the momentum vector has been replaced by p here,because, as in Section 5, we wish to consider p as the average of a particle momentum

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p. Only, here the average is conceived as taken over a statistical ensemble of possiblemomenta as well as over a ZBW period. So p = p(x) is a vector eld, and p = (x)p(x)is a momentum density, just as the Dirac current v = (x)v(x) is a particle density. Thevector part of (6.1) gives the so-called Gordon decomposition of the Dirac current

m0v = k + (Sei) (6.2)

where the Gordon currentk = (p cos q sin) (6.3)

is interpreted as a conduction current and Sei is said to be a spin density. But thisinterpretation is confounded by the peculiar role of . In particular, it seems inconsis-tent with the identication of S as spin density, which comes from angular momentumconservation.

A more sensible interpretation comes from rewriting (6.1) in the form

(p iq) = m0vei (S) + i( )S (6.4)

Its vector part,p = vm0 cos (S) + (iS) (6.5)

separates the momentum density p into a part which depends on the spin density Sand a part which does not. This equation should be compared with Eq. (5.9) for thetime-average momentum. The dierence between the two equations is presumably due toensemble averaging, so (5.9) should be recoverable by some kind of approximation to (6.5).Toward that end, we neglect the derivative of in

p = vm0 cos S + (vs) (6.6)

Next, to decouple Dirac streamlines the spatial variation between them must be ignored; inother words, assume that the variation of all quantities in (6.6) is along streamlines. Thisstreamline approximation can be expressed formally by writing

= vd

d(6.7)

in (6.6), so it becomesp = vm0 cos v S s (6.8)

Comparing this with (5.9), the rst thing to note is that the sign corresponding v S termis opposite. This discrepancy could be eliminated by changing a sign in the initial equation(5.3) determining the relation between p and S. However, Eq. (5.6), from which (5.9) isderived, is also derivable from the Dirac theory. Thus, the discrepancy is perplexing, but,with full condence in the internal consistency of the Dirac theory, let us presume that itcan be resolved and move on.

Comparison of (6.8) with (5.9) or (5.7) suggests

m = p v = m0 cos (6.9)

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In other words, the invariant cos describes a variation in mass. This interpretation isstrongly supported by a virial theorem derived from the Dirac equation in Ref. 5, whichproves that the entire energy shift of an electron in a Coulomb eld can be ascribed tocos.

The last term in (6.8) does not appear in (5.9), but it may indicate a need to generalizethe derivation of (5.9). It follows from (6.8) that

= p s1 (6.10)which could play a complementary role to (6.9).

By strict analogy with the reasoning that led to (5.12), to obtain (6.5) by averaging overZBW oscillations, we should presumably have

mr = (S) + (iS) (6.11)where the overbar now denotes an ensemble average as well as the time average in (5.12).

To complete our discussion of the Dirac equation (6.4), we will need to consider itstrivector part

q = vm0 sin + (Si) + S (6.12)One can obtain from this

q s = s2 (v) s(s v) (6.13a)and

q v = m0 sin + v s (s) (6.13b)Using the denition of q by (2.12) to evaluate the left side, these equations yield

(v) = 0 (6.14a)and

(s) = m0 sin (6.14b)The rst of these is the well-known conservation law for the Dirac current, and in theapproximation (6.7) it reduces to the identify v v = 0. The physical signicance of (6.14b)is dicult to divine. In the approximation (6.7) it reduces to

v s = m0 sin (6.15)which shouid be considered for a particle model along with (6.9) and (6.10). The mainissue is whether is an artifact of the averaging process or actually describes a real featureof the particle (such as variable mass). Of course, it could be a combination of the two.

Besides (6.13a,b), Eq. (6.12) implies two other relations which can be found by evaluatingq S. However, the result is contained in the dynamical relation which we consider next.

The dynamics of electron motion along a streamline in the Dirac theory is completelydeterminied by the rotational velocity , which can be calculated from the Dirac wavefunction. An explicit expression for in terms of observables was found in Ref. 3 to havethe form

m0 = eFei + Q + (m0 cos + ev A)S1 (6.16)

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where F is the external electromagnetic eld and Q is a complicated function of , , andS which need not be exhibited here. The overbar is to remind us that is to be regardedas an average in the sense specied above.

Inserting (6.16) into (2.18) and (2.19), we get equations of motion for the velocity andthe spin precession along a streamline:

m0v = (eFei + Q) v (6.17)

m0S = (eFei + Q) S (6.18)These are exact equations in the Dirac theory. Their most surprising feature is the -dependent duality rotation of the external eld F . This kind of rotation mixes electricand magnetic elds, so it may well be detectable experimentally, but there has been noeort to do so thus far. There seems to be nothing in the ZBW process to produce suchan eect, which again suggests that the peculiar parameter arises from the process ofaveraging.

The term Q v generalizes what Bohm has called the quantum force. It determinesthe bunching up and thinning out of streamlines at maxima and minima in diractionpatterns. If the speculations about ZBW elds in Section 4 are on track, this term mustdescribe statistical features of electron motion due to ZBW interactions. It may also includeself-interaction. Indeed, in the approximation (6.7) with = 0,

Q = ..S (6.19)

in agreement with (5.37), except for the sign. The consequences of this approximationhave been discussed at length in Ref. 5, with results essentially in agreement with those inSection 5.

REFERENCES

1. D. Hestenes, Quantum mechanics from self- interaction, Found. Phys. 15, 63-87(1985).

2. D. Hestenes, The Zitterbewegung interpretation of quantum mechanics, Found. Phys.20, 12131232 (1990). Further references are contained therein.

3. D. Hestenes, Local observables in the Dirac theory, J. Math. Phys. 14, 893905(1973).

4. D. Hestenes, Proper dynamics of a rigid point particle, J. Math. Phys. 15, 17781786(1974).

5. R. Gurtler and D. Hestenes, Consistency in the formulation of the Dirac, Pauli, andSchroedinger theories, J. Math. Phys. 16, 573583 (1975).

6. D. Hestenes, On decoupling probability from kinematics in quantum mechanics, inMaximum Entropy and Bayesian Methods (Dartmouth College, 1990) (Kluwer, Dor-drecht, 1991), pp. 161183.

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